I am in the process of transferring stock pursuant to a Will -

My signature must be guaranteed by a financial institution recognized by the company which issued the stock.

I've never seen this before.

Can someone explain why just a notary won't do and also is any bank (I would think) a financial institution, acceptable to the company?

Many financial institutions require a "gold medallion" signature guarantee, which must be provided by another financial institution. This is indeed different than a notary public. Not sure, but I believe that by doing this the institution that witnesses your signature is actually on the hook to reimburse the other institution for any losses that may occur due to fraud to a higher level than a notary public could. Whenever I've needed this type of signature guarantee I've gone to my local bank branch or stock broker. One advantage over a notary public: it's free.